اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدIndependence number of edge-chromatic critical graphs
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Let $G$ be a simple graph with maximum degree $Delta(G)$ and chromatic index $chi(G)$. A classic result of Vizing indicates that either $chi(G )=Delta(G)$ or $chi(G )=Delta(G)+1$. The graph $G$ is called $Delta$-critical if $G$ is connected, $chi(G )=Delta(G)+1$ and for any $ein E(G)$, $chi(G-e)=Delta(G)$. Let $G$ be an $n$-vertex $Delta$-critical graph. Vizing conjectured that $alpha(G)$, the independence number of $G$, is at most $frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $alpha(G)<frac{3n}{5}$. We show that for any given $varepsilonin (0,1)$, there exist positive constants $d_0(varepsilon)$ and $D_0(varepsilon)$ such that if $G$ is an $n$-vertex $Delta$-critical graph with minimum degree at least $d_0$ and maximum degree at least $D_0$, then $alpha(G)<(frac{{1}}{2}+varepsilon)n$. In particular, we show that if $G$ is an $n$-vertex $Delta$-critical graph with minimum degree at least $d$ and $Delta(G)ge (d+2)^{5d+10}$, then [ alpha(G) < left. begin{cases} frac{7n}{12}, & text{if $d= 3$; } frac{4n}{7}, & text{if $d= 4$; } frac{d+2+sqrt[3]{(d-1)d}}{2d+4+sqrt[3]{(d-1)d}}n<frac{4n}{7}, & text{if $dge 19$. } end{cases} right. ]
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